Field of the invention
-
The present invention relates a method for designing a flow device according to the
features of claim 1.
Prior Art
-
In conformity with the cited references, pages 23 and 24 of the present description.
1. Introduction and discussion of background
-
The present invention is related to the design and construction of flow devices and
especially turbine and compressor blade channels.
-
In recent years substantial efforts have been made to introduce inverse design
tools for airfoil, turbine blades and vane channel, and other flow devices for which the
optimum combination of aerodynamic forces and losses plays acrucial role. For an
adequate overview of the state of the art the reader is referred to work by Jameson et al.
[3], Giles and Drela [4], Dang and Isgro [5], Dulikravich [6], Pandya and Baysal [7]. An
overview of applications to current turbomachinery design is given by Jennions [8].
It should be understood that this list of publications on inverse design methods is far
from complete. Furthermore, it should be mentioned that formulations of the Euler
equations using two stream functions have been previously used (see Turner and Giles
[9], for example). However, with the exception of genuine inverse methods (Keller et
al. [10], [11]) that have been proposed for certain two-dimensional flows, the "inverse
methods" that are presently used to arrive at an optimum design of free-dimensional
flow devices are not really inverse. In general "Evolution Strategy", generic algorithms
or simply Newton's method are used to guide an extensive series of flow calculations
for gradually varying geometry toward some kind of optimum.
-
The key advantage of using inverse equations instead, is associated with the fact
that the geometry of a flow device does not have to be defined in advance, As an
alternative it is possible to impose the distribution of static pressure, flow speed,
temperature, heat flux or any other physical quantity, as a boundary condition and ask
for the geometry (as a past of the solution) that generates the desired distribution of
static pressure, etc.. Furthermore, if the original domain of definition is simply
connected, a one-to-one mapping into a cubic domain of definition with Cartesian
coordinates can be achieved. As a consequence, the generally tedious task of grid
generation for numerical calculations can be avoided.
2. Stream Functions and Natural Coordinate
-
To define boundary conditions and to embed boundary layers in an inviscid bulk flow,
for example, it may be useful to introduce the "natural coordinate" σ that is related to
the arc lengths along stream lines. Following the ideas of Ref. [2] we introduce a pair
of stream functions to define a vector potential for the velocity field. Without essential
loss of generality we restrict our considerations to solenoidal vector fields for the mass
flux. In order to extend the discussion presented in Ref. [2] to compressible flows the
mass flux (rather than the velocity) needs to be related to the stream functions. Hence,
introducing the two stream functions ψ and χ, we can write
where
u is the velocity vector and ρ the density. The natural coordinate can be
defined by
where σ is the dimensionless natural coordinate and
h = h(σ,ψ,χ) is a nonlinear
scretching coefficient, such that
hdσ denotes the actual length increment along the
streamlines, As usual the nonlinear stretching needs to be included, in order to reach
compatibility between the orthogonality requirement (2) and the uniform definition of
hdσ as the length increment along the streamlines.
-
Furthermore, in inverse coordinates we can write
where
x is the coordinate vector in the original space.
-
Fist we need to express the time-independent Euler equations in terms of the
two stream functions ψ and χ.
3. The Governing Equations in a Rotating Frame of Reference
-
An area of major interest to use inverse equations is the design of optimized turbine and
compressor blade channels. For this reason it seems to be appropriate to further extend
the governing equations to rotating frames of reference, In a frame of reference rotating
steadily at the angular velocity
Ω the Euler equation can be written in the form
where
p refers to the static pressure. The terms on the right-hand side of (4) contribute
to the fictitious volume force. The first term accounts for the Coriolis force and the
second for the centrifugal force. Making use of the identities
and
where
ω refers to the vorticity,
we can rewrite (4) as
-
The first law of thermodynamics can be expressed as
where
T is the temperature,
s denotes the entropy per unit mass and
cv refers to the
specific heat per unit mass at constant volume. The equation of state is
and
cp refers to the specific heat per unit mass at constant pressure. Combining (8),
(9) and (10) yields
-
Introducing the total enthalpy
equation (11) can now be expressed in the form of
Crocco's theorem in the rotating
frame:
-
By virtue of (1) the condition of mass conservation is identically satisfied. Moreover,
noting that the validity of the Euler equations is restricted to adiabatic flows, it is
obvious that the
s is constant on streamlines. Hence, the entropy is an integral of
motion (in any frame of reference),
-
Furthermore, it is obvious that
-
By virtue of (14) and (15) the scalar product of the velocity vector with equation (13)
yields
-
This is Bernoulli's theorem in the rotating frame of reference. It shows that the total
enthalpy in the rotating frame is an integral of motion. Making use of (1), (14) and
(16) equation (13) can be rewritten as
-
Furthermore, using the identity
the vector products of (17) with grad ψ and grad χ, respectively, lead to the following
generalized Euler equations in terms of two stream functions
-
For the subsequent discussion it is useful to introduce the mixed product that is defined
by
-
An aim of the present paper is to derive the inverse forms of Euler's equations, using
the two stream functions ψ and χ and the natural (streamline) coordinate σ as
independent variables.
4. Mapping to Inverse Coordinates
-
In this section we consider the mapping
-
The derivatives in the two coordinate systems are related by
where the mixed product
may be interpreted as Jacobi's determinant. Moreover it can be shown that
-
For a function
f( x ) = g(σ,ψ,χ) the conversion of derivatives can be expressed as
and for a vector
a ( x ) = b (σ,ψ,χ) we can write
-
Using (25), (26) and (28) it may be shown that
-
Similarly (or by analogy) we obtain
-
Furthermore, using (25) it may be shown that
-
Making use of (7), the identity
and (29) to (32) the equations of motion, (19) and (20), can be written as
-
Note that the mapping described in this section is thus far valid for any choice of
the variable σ. No use has been made yet of the fact that σ should ultimately be
interpreted as the natural coordinate as defined by (2) and (3).
-
The subsequent part of the analysis shall now account for the special
significance of σ as a natural coordinate. From (2), (3), (23) and (25) we obtain
-
Hence, (36) implies that the covariant and contravariant forms of
∂x/∂σ differ by the
factor
h2 . Furthermore, (36) implies
-
The first relation (37) is the same as (3). Using (37) the equations (34) and (35) can be
simplified to
-
It is quite remarkable that for irrotational isentropic flow,
equations (38) and (39) can be integrated once.
5. Thermodynamic Relations
-
Making use of (12) and the definitions
we can write
-
T o is the stagnation temperature in the absolute system,
c is the sound speed and
M
the Mach number in the rotating frame. Hence,
-
Combining (14) with the first law of thermodynamics and the equation of state we can
show that the isentropic relations
hold along streamlines. In these expressions
ρ o,
p o and
c o should be interpreted as
the stagnation density, the stagnation pressure and the stagnation sound speed,
respectively, in the absolute system and, by virtue of (14),
-
It is important to note that with the help of (14) and 44) to 46) all thermodynamic
quantities can be relaxed to integrals of motion! For the mass flux we obtain (see
(3) and the first equation (23))
-
We may now note thai the equations of motion (38), (39), together with the
condition (36), that defines the natural coordinate σ, and the thermodynamic
relations constitute a complete set of equations in inverse coordinates..
6. Cylindrical Polar Coordinates as Dependent Variables
-
For many applications, including flows through turbines, it is convenient to express the
coordinate vector
x in terms of cylindrical polar coordinates. Introducing cylindrical
polar coordinates (
x,
r,) it may be shown that Jacobi's determinant assumes the form
-
Assuming that the angular velocity vector
Ω is parallel to the axis and points in its
positive direction, we find
-
The relation (37), that defines the natural coordinate σ, may be expressed as
-
Inserting (48) to (50) in (38) and (39) leads to a complete set or equations in cylindrical
polar coordinates, if the equations (51), that define the natural coordinate, and the
thermodynamic relations are added.
7. Cartesian Coordinates as Dependent Variables
-
In most cases, including inverse design of wings, for example, the coordinate vector
x
is preferably expressed in terms of Cartesian coordinates and the frame of reference is
not rotating. Hence, we can set
Ω =
0. Introducing Cartesian coordinates (
x, y, z)
Jacobi's determinant assumes the form
-
The relation (37), that defines the natural coordinate σ, may be expressed as
8. Isentropic Flows
-
As pointed out earlier, the equations (38) and (39) can be substantially simplified if we
restrict the consideration to isentropic flows. In this case the conditions (40) lead to
where
f1 and
f 2 are functions of integration. Moreover, as we have not yet defined the
scaling of the natural coordinate σ the function
f(σ) can be "absorbed" by re-scaling
the natural coordinate,
-
For convenience we may also assume that the normalization of the density is included
in this stretching of the natural coordinate and from now on consider ρ as the density
that is normalized with the stagnation density. Hence we can write
-
On the other hand, by suitable scaling of the stream functions, the stagnation sound
speed can also be absorbed and (47) may be normalized accordingly,
-
Combining (36) and (56) yields
-
Eliminating
J and
h from (56), (58) and the first equation (37) leads to
-
We may now regard (59) as the equation for x = x (σ,ψ,χ), where the density ρ is
related to |∂ x /∂σ| by (57), (58) and (60). In other words, (57) and (58) may be
considered as a palametric representation (in terms of the Mach number) of the function
m = m(ρ) that can be inserted in (60) to obtain the desired expression for the density.
-
The structure of the equations becomes still more transparent by considering the
special case of plane two-dimensional flows. This simplification is achieved, for
example, by introducing Cartesian coordinates,
x = (x,y,z) and defining the stream
function χ by
-
In this case (59) can be reduced to
and (57), (58) and (60) can be put into the form
-
-
Combining (56), (58), (62) and (64) it may be shown that
-
Considering now an incremental strewn tube it is easy to show that the equations (62),
(63) and (65) are in agreement with the equations of motion. Eliminating ρdσ from
(62) and (63) we obtain the correct relation between massflow vector and velocity
vector. On the other hand (65) relates the massflow to the cross-sectional area of the
incremental stream tube. The massflow m reaches its maximum at M = 1. Hence, the
term on the right-hand side of (65) reaches its minimum at M = 1. Considering an
incremental Laval nozzle with slowly varying width dy along a line y = constant , for
example, the convergent-divergent character of the cross-section is immediately
apparent from (65). Nevertheless, it is surprising that the system of equations (62)
is generally elliptic. It appears that the hyperbolic character of supersonic flow is
hidden in the combination of (62) and (63) with ∂m/∂M < 0.
-
Finally, for the special case of incompressible flows (62) reduces to Laplace's
equation,
which is in agreement with the properties of a special type of conformal mapping that is
usually called hodograph transformation.
9. General Limitations with Respect to Supersonic Flow
-
An important limitation that applies to any coordinate inversion and is relevant for
supersonic flow appears if the Jacobian vanishes,
-
Flows that satisfy the condition (67) are not included in the solution space of the
inverse equations. The reader is referral to the corresponding discussion by Courant
and Friedriechs [1]. Flows that satisfy (67) are called simple waves. An expansion fan
in an isentropic flow, for example, is not a solution of the equations (57) to (60). To
obtain a complete set of solutions we need to add the solutions of (67) to the solution
space of any set of inverse equations. In Cartesian coordinates, for example, the
general solution of (67) can be expressed as
-
The validity of (68) is easily confirmed by introducing (52) in (67) and inserting (68).
10. Axisymmetric Flows
-
For the special case of axisymmetric flow in the absolute frame of reference the
equations (19) and (20) reduce to a generalized form of the Bragg-Hawthorne equation.
Without giving the details of deriving the equation we simply state the results. For
axisymmetric flows we introduce a single stream function ψ and the specific angular
momentum
C as an additional integral of motion:
where
x and
r are the axial and radial coordinates, and
u,
v and
w denote the axial,
radial and azimuthal velocity components, respectively. In this case the Euler equation
is found to be
where the total enthalpy
and the entropy
s = s(ψ) are again integrals of motion. Equation (47) reduces to
-
The remaining equations are the same as in section 5. We may note that, regarding the
density ρ and the entropy
s as constants, replacing the total enthalpy (71) by the total
head,
and eliminating the density from the definition of the stream function,
(70) reduces to the well-known Bragg-Hawthorne equation for incompressible
axisymmetric flow:
-
At this point we introduce the mapping
to inverse coordinates and, for simplification, we restrict the consideration to
incompressible flow. In this case the streamline coordinate σ is defined by
where
h = h(σ,ψ) is the stretching function that provides the basis for compatibility
between orthogonality of the curves σ(
x,r) = constant and ψ(
x,
r) = constant and the
uniform meaning of
hdσ,
as the length increment along streamlines. The condition for the curvilinear coordinates
to be orthogonal is again
-
Introducing Jacobi's determinant,
the partial derivatives σ(
x,
r) and ψ(
x,
r) can be expressed as
-
For a function
f(x,r) = g(σ,ψ) the conversion of derivatives can be expressed as
-
Inserting (81) in (82) we obtain the conversion of the second derivatives;
-
Adding the second derivatives according to (83), and making use of (78), (79) and
(80), integration by parts yields
-
Furthermore, from (78), (79) and (80) it follows that
-
Combining (75) with (84) and (85) we finally obtain the inverse Bragg-Hawthorne
equation
-
A complete system of equations is obtained by adding (78) and (79). The equation (86)
is again remarkably simple. To check the previous results a somewhat more elaborate
analysis has been chosen in this case to derive the inverse equation, It is tedious, but
rather straightforward to eliminate the Jacobian from (86) in favor of partial derivatives
of the radius. Making use of (78), (79) and (80) the equation (86) can be written in the
form
and the derivative of the coordinate stretching factor with respect to the stream function
can be expressed as
-
The idea is now to solve (87) by means of a Gauss-Seidel iteration scheme and
determine the coordinate stretching factor
h using a stepwise integration of (88). From
(74), (78), (81) and (85) we obtain
-
The flow speed can now it expressed as
11. Difficulties Encountered with Applications of Inverse Equations
-
For the transform of the equations to their inverse form it is essential to choose the
domain of definition such that a one-to-one mapping is generated, For simply
connected domains of definition, such as turbine blade channels, for example, this
difficulty would generally not appear. However, in the case of flow fields that are not
simply connected the difficulty arises that the entire flow field must be suitably cut to
obtain domains for which a one-to-one mapping exists. A further point that should be
addressed concerns the chosen domain of definition for the stream functions. In this
context the reader is referred to Keller [12] for more details. If a wake appears within
the domain of definition there may be a problem with streamlines, originating on the
downstream side, for which the integrals of motion have not been defined. For this
reason it is a good idea to include a check in a numerical solution procedure that warns
the user in cases where integrals of motion have been taken beyond their intended
domain of definition.
12. Two Simple Examples
-
To demonstrate the capabilities of inverse equations we consider two simple
axisymmetric problems. The design of a compressor diffuser of a gas turbine is often
strongly restricted by the space requirements of turbine and combustor. As a
consequence there is usually the necessity of choosing a curved diffuser. Although
there is broad empirical knowledge on the performance of straight annular diffusers
(see Kline, Abbott and Fox [13], for example), it may often be difficult to apply this
information to arbitrary curved diffusers. However, knowing the relation between the
largest acceptable adverse pressure gradient and the ratio of (local) length and inlet
width of a diffuser we can specify restrictions for the static pressure distributions along
the inner and outer walls of a curved annular diffuser. As indicated in figure 1, a
possible strategy to design a curved diffuser may be defined as follows. Near the
diffuser inlet the flow speeds along both walls first follow a curve the corresponds to
the largest acceptable pressure gradient. After a certain arc length a smooth transition to
a different curve of maximum pressure increase is chosen for the outer diffuser wall.
For a certain distance the flow speed distributions along the two walls follow two
different curves of largest pressure increase. This domain will generate the curved
portion of the diffuser. Toward the exit of the diffuser a smooth transition of the flow
speed along the inner wall is introduced, in order to obtain a velocity profile at the
diffuser exit that is as uniform as possible. The second transition brings the flow speed
along the inner wall back to a common curve of maximum pressure increase. Knowing
the flow speed distributions along both diffuser walls enables us to express ∂r/∂σ in
terms of ∂r/∂ψ, making use of equation (90). At the inlet to the diffuser we set
∂r/∂σ = 0, and at the exit of the diffuser we choose a linear transition for ∂r/∂σ
between the corresponding values at the endpoints of the two walls. Choosing any
kind of distributions for the total head and the circulation at the diffuser inlet does not
cause additional difficulties, because H = H(ψ) and C = C(ψ) appear as integrals of
motion in the governing equations (87) and (88). It is now easy to find a numerical
solution with the help of the Gauss-Seidel iteration scheme. For more details the reader
is referred to Keller et al. [10]. For the simple example presented it has been assumed
that the flow is incompressible, the local head is uniform and the flow is free of swirl.
Figure 2 illustrates a streamline map that corresponds to the boundary conditions
defined by the flow speed distributions shown in figure 1. Furthermore, figure 2
shows lines of constant σ. It should be pointed out that, according to the definition of
the natural coordinate, lines of constant σ are always orthogonal to streamlines (i.e.,
lines of constant ψ and χ). Thus we may observe that one of the remarkable spin-offs
of using inverse equations is the automatic generation of an orthogonal contour-fitted
coordinate system in the original space. Although the actual computation is done
in a cubic domain of definition with Cartesian coordinates in inverse space, a
curvilinear orthogonal grid is simultaneously generated together with the solution in the
original space. Furthermore, the density of grid lines gets automatically high in
domains of large gradients. For these reasons the accuracy of the method is remarkably
high (for a fixed number of grid points) and the convergence is very fast.
-
The second example (see figure 3) is a turbine inlet flow. In this case the
geometry of an annular duct is imposed as a boundary condition, combined with a
linear variation of the flow angles at the inlet and exit, respectively. According to
ABB's convention the flow is from right to left. This is one of the examples that has
been used to check the equations. To include and discuss three-dimensional examples
of inverse design would exceed the scope of this paper. Such results will be presented
elsewhere.
13. Concluding Remarks
-
With the transformation of the Euler equations to their inverse form, using a pair of
stream functions and the natural coordinate as independent variables, a theoretical basis
has been introduced for a broad class of extremely powerful inverse computation
methods. Instead of searching for a geometry of a flow device that will lead to certain
physical properties, the physical properties can be defined in advance and the solution
of the inverse equations directly leads to the corresponding geometry. It has become
easier to arrive at an optimized design. However, the difficulty of defining an optimum
remains.
-
From a practical point of view it may be convenient to proceed as follows. As a
preparation for the optimum design of a certain flow device we introduce a first guess
for its geometry, that may satisfy certain basic constraints. Secondly we use the
inverse method for a "backward computation". In other words, we specify the
geometry according to the first guess and determine the static pressure distribution
along the surface. In a second step we modify the static pressure distribution along the
surface according to the requirements or the optimization and introduce the modified
static pressure distribution as the boundary condition for the "forward computation". It
should be noted that the inverse equations provide an excellent basis for using both
physical quantities or the geometry of a flow device to define the boundary conditions.
In fact it is even easy to use combinations of geometrical and physical boundary
conditions.
-
Hence, it seems that there is justified hope that computation methods that are
based on inverse equations will become extremely efficient tools to arrive at optimized
flow devices.
-
Although it might appear that the existence of integrals of motion is essential to
derive a set of antisymmetric equations of the form (19), (29). However, it is
interesting that the extension to non-adiabatic flows of both antisymmetric equations
and transformation to inverse coordinates is relatively simple, including the case of the
general time-independent Navier-Stokes equations.
References
-
- [1] R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience,
New York (1967).
- [2] J. J. Keller, A pair of stream functions for three-dimensional vortex flows,
Z. angew. Math. Phys. 47(1996), 821.
- [3] A. Jameson, N. A. Pierce, and L. Martinelli, Optimum aerodynamic design using
the Navier-Stokes equations. AIAA paper 97-0101, 35th Aerospace Sciences
Meeting, Reno, Nevada 1997.
- [4] M. B. Giles and M. Drela, Two-dimensional transonic aerodynamic design method,
AIAA Journal 25(1987), 1199
- [5] T. Dang and V. Isgro, Inverse method for turbomachine blades using existing time-marching
techniques, ASME paper 94-GT-20, International Gas Turbine and
Aeroengine Congress, The Hague, Netherlands, 1994.
- [6] G. S. Dulikravich, Shape inverse design and optimization for three-dimensional
aerodynamics, AIAA paper 95-0695, 33rd Aerospace Sciences Meeting, Reno,
Nevada 1995.
- [7] M. J. Pandya and O. Baysal, Gradient-based aerodynamic shape optimization using
ADI method for large-scale problems, AIAA paper 96-0091, 34th Aerospace
Sciences Meeting, Reno, Nevada 1996.
- [8] I. K. Jennions, The role of CFD in the design process, AGARD Paper, 7 Rue
Ancelle, 92200 Neuilly sur Seine, France, Lecture Series 195, May to June
1994.
- [9] M. G. Turner and M. B. Giles, Design and analysis of internal flow fields using a
two stream function formulation, ASME Winter Annual Meeting 1990, FED
Vol. 103, 203.
- [10] J. J. Keller, W. Egli, and J. Exley, Force- and loss-free transitions between flow
states, Z. angew. Math. Phys. 36(1985), 854.
- [11] J. J. Keller, W. Egli, and R. Althaus, Vortex breakdown as a fundamental element
of vortex dynamics, Z. angew. Math. Phys. 39(1988), 404.
- [12] J. J. Keller, On the interpretation of vortex breakdown, Phys Fluids 7 (7),
(1995), 1696.
- [13] S. J. Kline, D. E. Abbott and R. W. Fox, Optimum design of straight-walled
diffusers, J. Basic Eng., Trans. ASME81, Series D (Sept 1959).
-
Figure Captions
-
- Figure 1:
- Distributions of flow speed along the two walls of an annular diffuser. The
solid lines mark curves of maximum adverse pressure gradient. Smooth
transitions between these curves have been generated using forth-order
polynomials, in order to define suitable distributions of flow speed along the
diffuser walls.
- Figure 2:
- Map of streamlines and curves of constant σ that corresponds to the
boundary conditions defined by the distributions of flow speed shown in
figure 1.
- Figure 3::
- Map of streamlines and curves of constant σ of a turbine inlet flow,
where σ is the dimensionless natural coordinate and
and
where ω refers to the vorticity,
we can rewrite (4) as
and cp refers to the specific heat per unit mass at constant pressure. Combining (8), (9) and (10) yields
may be interpreted as Jacobi's determinant. Moreover it can be shown that
where the total enthalpy
and the entropy
(70) reduces to the well-known Bragg-Hawthorne equation for incompressible axisymmetric flow:
where
as the length increment along streamlines. The condition for the curvilinear coordinates to be orthogonal is again
and wherein h is the stretching factor of the natural coordinate defined by
with the auxiliary equations
wherein
